Potential landscape of high dimensional nonlinear stochastic dynamics with large noise

Proliferation of unstable states and their impact on stochastic out-of-equilibrium dynamics in two coupled Kerr parametric oscillators

Networks of nonlinear parametric resonators are promising candidates as Ising machines for annealing and optimization. These many-body out-of-equilibrium systems host complex phase diagrams of coexisting stationary states. The plethora of states manifest via a series of bifurcations, including bifurcations that proliferate purely unstable solutions. Here we demonstrate that the latter take a fundamental role in the stochastic dynamics of the system. Specifically, they determine the switching paths and the switching rates between stable solutions. We demonstrate experimentally the impact of the added unstable states on noise-activated switching dynamics in a network of two coupled parametric resonators.

Mapping transcriptomic vector fields of single cells

Single-cell (sc)-RNA-seq, together with RNA-velocity and metabolic labeling, reveals cellular states and transitions at unprecedented resolution. Fully exploiting these data, however, requires kinetic models capable of unveiling governing regulatory functions. Here, we introduce an analytical framework dynamo, that infers absolute RNA velocity, reconstructs continuous vector-field functions that predict cell fates, employs differential geometry to extract underlying regulations, and ultimately predicts optimal reprogramming paths and perturbation outcomes. We highlight dynamo’s power to overcome fundamental limitations of conventional splicing-based RNA velocity analyses to enable accurate velocity estimations on a metabolically-labeled human hematopoiesis scRNA-seq dataset. Furthermore, differential geometry analyses reveal mechanisms driving early megakaryocyte appearance and elucidate asymmetrical regulation within the PU.1–GATA1 circuit. Leveraging the Least-Action-Path method, dynamo accurately predicts drivers of numerous hematopoietic transitions. Finally, in silico perturbations predict cell-fate diversions induced by gene perturbations. Dynamo thus represents an important step in advancing quantitative and predictive theories of cell-state transitions.

Can Systems Biology Advance Clinical Precision Oncology?

Precision oncology is perceived as a way forward to treat individual cancer patients. However, knowing particular cancer mutations is not enough for optimal therapeutic treatment, because cancer genotype-phenotype relationships are nonlinear and dynamic. Systems biology studies the biological processes at the systems' level, using an array of techniques, ranging from statistical methods to network reconstruction and analysis, to mathematical modeling. Its goal is to reconstruct the complex and often counterintuitive dynamic behavior of biological systems and quantitatively predict their responses to environmental perturbations. In this paper, we review the impact of systems biology on precision oncology. We show examples of how the analysis of signal transduction networks allows to dissect resistance to targeted therapies and inform the choice of combinations of targeted drugs based on tumor molecular alterations. Patient-specific biomarkers based on dynamical models of signaling networks can have a greater prognostic value than conventional biomarkers. These examples support systems biology models as valuable tools to advance clinical and translational oncological research.

The complex landscape of haematopoietic lineage commitments is encoded in the coarse-grained endogenous network

Haematopoietic lineage commitments are presented by a canonical roadmap in which haematopoietic stem cells or multipotent progenitors (MPPs) bifurcate into progenitors of more restricted lineages and ultimately mature to terminally differentiated cells. Although transcription factors playing significant roles in cell-fate commitments have been extensively studied, integrating such knowledge into the dynamic models to understand the underlying biological mechanism remains challenging. The hypothesis and modelling approach of the endogenous network has been developed previously and tested in various biological processes and is used in the present study of haematopoietic lineage commitments. The endogenous network is constructed based on the key transcription factors and their interactions that determine haematopoietic cell-fate decisions at each lineage branchpoint. We demonstrate that the process of haematopoietic lineage commitments can be reproduced from the landscape which orchestrates robust states of network dynamics and their transitions. Furthermore, some non-trivial characteristics are unveiled in the dynamical model. Our model also predicted previously under-represented regulatory interactions and heterogeneous MPP states by which distinct differentiation routes are intermediated. Moreover, network perturbations resulting in state transitions indicate the effects of ectopic gene expression on cellular reprogrammes. This study provides a predictive model to integrate experimental data and uncover the possible regulatory mechanism of haematopoietic lineage commitments.

Noise distorts the epigenetic landscape and shapes cell-fate decisions

The Waddington epigenetic landscape has become an iconic representation of the cellular differentiation process. Recent single-cell transcriptomic data provide new opportunities for quantifying this originally conceptual tool, offering insight into the gene regulatory networks underlying cellular development. While many methods for constructing the landscape have been proposed, by far the most commonly employed approach is based on computing the landscape as the negative logarithm of the steady-state probability distribution. Here, we use simple models to highlight the complexities and limitations that arise when reconstructing the potential landscape in the presence of stochastic fluctuations. We consider how the landscape changes in accordance with different stochastic systems and show that it is the subtle interplay between the deterministic and stochastic components of the system that ultimately shapes the landscape. We further discuss how the presence of noise has important implications for the identifiability of the regulatory dynamics from experimental data. A record of this paper's transparent peer review process is included in the supplemental information.

Machine learning framework for computing the most probable paths of stochastic dynamical systems

The emergence of transition phenomena between metastable states induced by noise plays a fundamental role in a broad range of nonlinear systems. The computation of the most probable paths is a key issue to understanding the mechanism of transition behaviors. The shooting method is a common technique for this purpose to solve the Euler-Lagrange equation for the associated action functional, while losing its efficacy in high-dimensional systems. In the present work, we develop a machine learning framework to compute the most probable paths in the sense of Onsager-Machlup action functional theory. Specifically, we reformulate the boundary value problem of a Hamiltonian system and design a neural network to remedy the shortcomings of the shooting method. The successful applications of our algorithms to several prototypical examples demonstrate its efficacy and accuracy for stochastic systems with both (Gaussian) Brownian noise and (non-Gaussian) Lévy noise. This approach is effective in exploring the internal mechanisms of rare events triggered by random fluctuations in various scientific fields.

Modeling Basins of Attraction for Breast Cancer Using Hopfield Networks

Cancer is a genetic disease for which traditional treatments cause harmful side effects. After two decades of genomics technological breakthroughs, personalized medicine is being used to improve treatment outcomes and mitigate side effects. In mathematical modeling, it has been proposed that cancer matches an attractor in Waddington's epigenetic landscape. The use of Hopfield networks is an attractive modeling approach because it requires neither previous biological knowledge about protein-protein interactions nor kinetic parameters. In this report, Hopfield network modeling was used to analyze bulk RNA-Seq data of paired breast tumor and control samples from 70 patients. We characterized the control and tumor attractors with respect to their size and potential energy and correlated the Euclidean distances between the tumor samples and the control attractor with their corresponding clinical data. In addition, we developed a protocol that outlines the key genes involved in tumor state stability. We found that the tumor basin of attraction is larger than that of the control and that tumor samples are associated with a more substantial negative energy than control samples, which is in agreement with previous reports. Moreover, we found a negative correlation between the Euclidean distances from tumor samples to the control attractor and patient overall survival. The ascending order of each node's density in the weight matrix and the descending order of the number of patients that have the target active only in the tumor sample were the parameters that withdrew more tumor samples from the tumor basin of attraction with fewer gene inhibitions. The combinations of therapeutic targets were specific to each patient. We performed an initial validation through simulation of trastuzumab treatment effects in HER2+ breast cancer samples. For that, we built an energy landscape composed of single-cell and bulk RNA-Seq data from trastuzumab-treated and non-treated HER2+ samples. The trajectory from the non-treated bulk sample toward the treated bulk sample was inferred through the perturbation of differentially expressed genes between these samples. Among them, we characterized key genes involved in the trastuzumab response according to the literature.

Climbing Escher's stairs: A way to approximate stability landscapes in multidimensional systems

Stability landscapes are useful for understanding the properties of dynamical systems. These landscapes can be calculated from the system's dynamical equations using the physical concept of scalar potential. Unfortunately, it is well known that for most systems with two or more state variables such potentials do not exist. Here we use an analogy with art to provide an accessible explanation of why this happens and briefly review some of the possible alternatives. Additionally, we introduce a novel and simple computational tool that implements one of those solutions: the decomposition of the differential equations into a gradient term, that has an associated potential, and a non-gradient term, that lacks it. In regions of the state space where the magnitude of the non-gradient term is small compared to the gradient part, we use the gradient term to approximate the potential as quasi-potential. The non-gradient to gradient ratio can be used to estimate the local error introduced by our approximation. Both the algorithm and a ready-to-use implementation in the form of an R package are provided.

Adaptive Landscape Shaped by Core Endogenous Network Coordinates Complex Early Progenitor Fate Commitments in Embryonic Pancreas

The classical development hierarchy of pancreatic cell fate commitments describes that multipotent progenitors (MPs) first bifurcate into tip cells and trunk cells, and then these cells give rise to acinar cells and endocrine/ductal cells separately. However, lineage tracings reveal that pancreatic progenitors are highly heterogeneous in tip and trunk domains in embryonic pancreas. The progenitor fate commitments from multipotency to unipotency during early pancreas development is insufficiently characterized. In pursuing a mechanistic understanding of the complexity in progenitor fate commitments, we construct a core endogenous network for pancreatic lineage decisions based on genetic regulations and quantified its intrinsic dynamic properties using dynamic modeling. The dynamics reveal a developmental landscape with high complexity that has not been clarified. Not only well-characterized pancreatic cells are reproduced, but also previously unrecognized progenitors-tip progenitor (TiP), trunk progenitor (TrP), later endocrine progenitor (LEP), and acinar progenitors (AciP/AciP2) are predicted. Further analyses show that TrP and LEP mediate endocrine lineage maturation, while TiP, AciP, AciP2 and TrP mediate acinar and ductal lineage maturation. The predicted cell fate commitments are validated by analyzing single-cell RNA sequencing (scRNA-seq) data. Significantly, this is the first time that a redefined hierarchy with detailed early pancreatic progenitor fate commitment is obtained.

Stochastic resonance for nonequilibrium systems

Stochastic resonance (SR) is a prominent phenomenon in many natural and engineered noisy systems, whereby the response to a periodic forcing is greatly amplified when the intensity of the noise is tuned to within a specific range of values. We propose here a general mathematical framework based on large deviation theory and, specifically, on the theory of quasipotentials, for describing SR in noisy N-dimensional nonequilibrium systems possessing two metastable states and undergoing a periodically modulated forcing. The drift and the volatility fields of the equations of motion can be fairly general, and the competing attractors of the deterministic dynamics and the edge state living on the basin boundary can, in principle, feature chaotic dynamics. Similarly, the perturbation field of the forcing can be fairly general. Our approach is able to recover as special cases the classical results previously presented in the literature for systems obeying detailed balance and allows for expressing the parameters describing SR and the statistics of residence times in the two-state approximation in terms of the unperturbed drift field, the volatility field, and the perturbation field. We clarify which specific properties of the forcing are relevant for amplifying or suppressing SR in a system and classify forcings according to classes of equivalence. Our results indicate a route for a detailed understanding of SR in rather general systems.

Discrete and continuous models of probability flux of switching dynamics: Uncovering stochastic oscillations in a toggle-switch system

The probability flux and velocity in stochastic reaction networks can help in characterizing dynamic changes in probability landscapes of these networks. Here, we study the behavior of three different models of probability flux, namely, the discrete flux model, the Fokker-Planck model, and a new continuum model of the Liouville flux. We compare these fluxes that are formulated based on, respectively, the chemical master equation, the stochastic differential equation, and the ordinary differential equation. We examine similarities and differences among these models at the nonequilibrium steady state for the toggle switch network under different binding and unbinding conditions. Our results show that at a strong stochastic condition of weak promoter binding, continuum models of Fokker-Planck and Liouville fluxes deviate significantly from the discrete flux model. Furthermore, we report the discovery of stochastic oscillation in the toggle-switch system occurring at weak binding conditions, a phenomenon captured only by the discrete flux model.

Transitions across Melancholia States in a Climate Model: Reconciling the Deterministic and Stochastic Points of View

The Earth is well known to be, in the current astronomical configuration, in a regime where two asymptotic states can be realized. The warm state we live in is in competition with the ice-covered snowball state. The bistability exists as a result of the positive ice-albedo feedback. In a previous investigation performed on a intermediate complexity climate model we identified the unstable climate states (melancholia states) separating the coexisting climates, and studied their dynamical and geometrical properties. The melancholia states are ice covered up to the midlatitudes and attract trajectories initialized on the basin boundary. In this Letter, we study how stochastically perturbing the parameter controlling the intensity of the incoming solar radiation impacts the stability of the climate. We detect transitions between the warm and the snowball state and analyze in detail the properties of the noise-induced escapes from the corresponding basins of attraction. We determine the most probable paths for the transitions and find evidence that the melancholia states act as gateways, similarly to saddle points in an energy landscape.

Discrete flux and velocity fields of probability and their global maps in reaction systems

Stochasticity plays important roles in reaction systems. Vector fields of probability flux and velocity characterize time-varying and steady-state properties of these systems, including high probability paths, barriers, checkpoints among different stable regions, as well as mechanisms of dynamic switching among them. However, conventional fluxes on continuous space are ill-defined and are problematic when at the boundaries of the state space or when copy numbers are small. By re-defining the derivative and divergence operators based on the discrete nature of reactions, we introduce new formulations of discrete fluxes. Our flux model fully accounts for the discreetness of both the state space and the jump processes of reactions. The reactional discrete flux satisfies the continuity equation and describes the behavior of the system evolving along directions of reactions. The species discrete flux directly describes the dynamic behavior in the state space of the reactants such as the transfer of probability mass. With the relationship between these two fluxes specified, we show how to construct time-evolving and steady-state global flow-maps of probability flux and velocity in the directions of every species at every microstate and how they are related to the outflow and inflow of probability fluxes when tracing out reaction trajectories. We also describe how to impose proper conditions enabling exact quantification of flux and velocity in the boundary regions, without the difficulty of enforcing artificial reflecting conditions. We illustrate the computation of probability flux and velocity using three model systems, namely, the birth-death process, the bistable Schlögl model, and the oscillating Schnakenberg model.

Escape rate for nonequilibrium processes dominated by strong non-detailed balance force

Quantifying the escape rate from a meta-stable state is essential to understand a wide range of dynamical processes. Kramers' classical rate formula is the product of an exponential function of the potential barrier height and a pre-factor related to the friction coefficient. Although many applications of the rate formula focused on the exponential term, the prefactor can have a significant effect on the escape rate in certain parameter regions, such as the overdamped limit and the underdamped limit. There have been continuous interests to understand the effect of non-detailed balance on the escape rate; however, how the prefactor behaves under strong non-detailed balance force remains elusive. In this work, we find that the escape rate formula has a vanishing prefactor with decreasing friction strength under the strong non-detailed balance limit. We both obtain analytical solutions in specific examples and provide a derivation for more general cases. We further verify the result by simulations and propose a testable experimental system of a charged Brownian particle in electromagnetic field. Our study demonstrates that a special care is required to estimate the effect of prefactor on the escape rate when non-detailed balance force dominates.

Reversed graph embedding resolves complex single-cell trajectories

Single-cell trajectories can unveil how gene regulation governs cell fate decisions. However, learning the structure of complex trajectories with multiple branches remains a challenging computational problem. We present Monocle 2, an algorithm that uses reversed graph embedding to describe multiple fate decisions in a fully unsupervised manner. We applied Monocle 2 to two studies of blood development and found that mutations in the genes encoding key lineage transcription factors divert cells to alternative fates.

Reversed graph embedding resolves complex single-cell trajectories

Single-cell trajectories can unveil how gene regulation governs cell fate decisions. However, learning the structure of complex trajectories with multiple branches remains a challenging computational problem. We present Monocle 2, an algorithm that uses reversed graph embedding to describe multiple fate decisions in a fully unsupervised manner. We applied Monocle 2 to two studies of blood development and found that mutations in the genes encoding key lineage transcription factors divert cells to alternative fates.